Let T be an operator from C^1(R) into C(R) satisfying the chain rule functional equation T ( f o g) = (T f) o g * (T g) . We show that in a non-degenerate case any solution of this equation has the form (T f)(x) = H (f(x)) / H(x)* |f'(x)|^p * {sgn(f'(x)) , where H is continuous and p > 0 and where the last term sgn(f'(x)) may be missing; then also p = 0 is possible. An "initial condition" like T(2*Id) = 2 will imply that T f = f' holds. We also consider T operating on smoother functions C^k(R) or C^inf(R) and n-dimensional generalizations of the chain rule equation.